Graph, clique and facet of boolean logical polytope

Logical analysis of data (LAD) discovers useful knowledge from a set of data in the form of a Boolean pattern for classifying future data. Generating a pattern has been shown to be equivalent to solving a 0–1 multilinear program (MP). Thus, the success of LAD is tightly related to how efficiently practical instances of pattern generation MP’s can be solved. For a polyhedral relaxation of LAD pattern generation MP, this paper introduces a new notion of similarity among data that allows for simultaneously relaxing multiple terms of the objective function of MP into a single valid inequality for the Boolean MP polytope. Specifically, we present a framework for constructing three types of strong valid inequalities from cliques in multiple graph representations of data that collectively yield a tight polyhedral relaxation of MP. Furthermore, we specify conditions under which each type of the new inequalities defines a facet of the MP polytope. In comparison with methods from the literature, benefits of the new inequalities are validated through classification experiments with 8 public machine learning datasets.